JHEP11(2013)004

Published for SISSA by Springer

Received: September 16, 2013

Accepted: October 18, 2013

Published: November 4, 2013

Supersymmetric defect models and mirror symmetry

Anson Hook,a Shamit Kachrub and Gonzalo Torrobab

aSchool of Natural Sciences, Institute for Advanced Study,

Princeton, NJ 08540, U.S.A.bStanford Institute for Theoretical Physics, Department of Physics and Theory Group, SLAC,

Stanford University, Stanford, CA 94305, U.S.A.

E-mail: hook@ias.edu, skachru@stanford.edu, torrobag@stanford.edu

Abstract: We study supersymmetric field theories in three space-time dimensions doped

by various configurations of electric charges or magnetic fluxes. These are supersymmetric

avatars of impurity models. In the presence of additional sources such configurations are

shown to preserve half of the supersymmetries. Mirror symmetry relates the two sets of

configurations. We discuss the implications for impurity models in 3d N = 4 QED with a

single charged hypermultiplet (and its mirror, the theory of a free hypermultiplet) as well as

3d N = 2 QED with one flavor and its dual, a supersymmetric Wilson-Fisher fixed point.

Mirror symmetry allows us to find backreacted solutions for arbitrary arrays of defects in

the IR limit of N = 4 QED. Our analysis, complemented with appropriate string theory

brane constructions, sheds light on various aspects of mirror symmetry, the map between

particles and vortices and the emergence of ground state entropy in QED at finite density.

Keywords: Supersymmetry and Duality, Supersymmetric gauge theory, Duality in Gauge

Field Theories

ArXiv ePrint: 1308.4416

Open Access doi:10.1007/JHEP11(2013)004

JHEP11(2013)004

Contents

1 Introduction 1

2 Three dimensional theories and mirror symmetry 2

2.1 Three-dimensional supersymmetric theories 3

2.2 Mirror symmetry for N = 4 theories 5

2.3 Mirror symmetry for N = 2 theories 7

3 SUSY defects and mirror symmetry in N = 4 theories 8

3.1 Adding electric charges to N = 4 SQED 8

3.1.1 Classical solutions and field of a point charge 10

3.2 Global vortices in the mirror configuration 11

3.2.1 Classical solutions and external vortices 13

3.2.2 Mapping to the electric variables 14

3.3 Adding magnetic charges to N = 4 SQED 15

3.4 Finite chemical potential in the mirror dual 17

4 SUSY defects and mirror symmetry in N = 2 theories 18

4.1 Semiclassical solutions 18

4.2 Comments on quantum dynamics 19

5 Lessons for impurity physics 19

6 Non-supersymmetric deformations and ground-state entropy 22

7 D-brane picture 23

7.1 Engineering the field theories 23

7.2 Including the background sources 24

8 Discussion and future directions 26

1 Introduction

Duality is a powerful tool in analyzing quantum field theories. An early and surprising

manifestation was the discovery of the relationship between the XY model and the abelian

Higgs model in 2+1 dimensions [1, 2]. This duality has been generalized to models with

additional ‘flavors’ of matter fields in [3, 4], and plays a role in the study of lattice models

of antiferromagnets.

The study of these field theories in the presence of external charges (e.g., impurities)

is of considerable interest. A single defect interacting with a Wilson-Fisher critical theory

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JHEP11(2013)004

was discussed in [5]. In general it is a tough question to determine the backreaction of

a given configuration of defects on the bulk field theory, and even tougher to compute

quantities of interest for IR transport, like current correlators in a lattice of defects.

Here, we focus on supersymmetric field theories. Theories with 3d N = 4 and N = 2

supersymmetry were shown to enjoy a mirror symmetry [6–11], where two distinct UV

gauge theories flow to the same IR fixed point. Mirror symmetry is in many ways a super-

symmetric cousin of the XY/abelian Higgs duality [12], and so many of the questions of

interest in that context can be imported to 3d supersymmetric mirror pairs. The additional

theoretical tools afforded by supersymmetry allow some of these questions to be answered.

In this paper, we focus on questions of charged defect or impurity physics in these

supersymmetric theories; earlier work in this spirit, where the focus was on holographic

supersymmetric constructions, includes [13–17]. We study both electric and magnetic

impurities in the simplest mirror pairs (reviewed in section 2) of theories with N = 4 and

N = 2 supersymmetry. We will see in sections 3 and 4 that such impurities can preserve 1/2

of the supersymmetry in the presence of appropriate external backgrounds for additional

fields. In particular, we will show that it is possible to preserve supersymmetry at finite

density for local and global U(1) symmetries.

We then find that the power of mirror symmetry allows us to extract non-trivial in-

formation about the IR nature of the solution for bulk fields in the presence of defects,

and even (in the original N = 4 example) allows us to compute in section 5 the lattice

backreaction at strong coupling. This gives a promising method for studying lattices of

impurities interacting with strongly coupled field theories. Besides applications to such

systems, our results contribute to the understanding of various formal aspects of mirror

symmetry, and provide a more explicit map between particles and vortices.

At some points it is useful to make contact with string theory. Various questions that

have arisen in studies of holography (such as the finite ground-state entropy of certain

doped field theories) can be viewed in a different light in our constructions, along the lines

envisioned in [18–20]. In section 6 we show that our construction explains the emergence

of ground-state degeneracy in strongly interacting QFTs at finite density. Specifically,

the N = 4 SQED theory at finite chemical potential for the topological U(1) symmetry

(defined in (2.7) below) is equivalent in the IR to free electrons in an external magnetic field.

The ground state entropy comes from the Landau level degeneracy of the dual. On the

other hand, the construction of 3d supersymmetric gauge theories via brane configurations

(following [21]) makes manifest many of the properties of mirror symmetry in the presence

of defects. This is studied in section 7. Finally, section 8 suggests various future directions

motivated by our results.

2 Three dimensional theories and mirror symmetry

Here, we discuss the field content and Lagrangians of the theories we’ll be interested in

throughout the rest of the paper. These are three-dimensional field theories with N = 2

and N = 4 supersymmetry, namely 4 and 8 supercharges respectively. We will do this in

terms of N = 2 multiplets, since these follow from dimensional reduction of the familiar

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JHEP11(2013)004

4d N = 1 multiplets. Theories with N = 2 have simpler matter content than their N = 4

counterparts, but their infrared dynamics is richer and more involved. For this reason, we

will first consider N = 4 theories. We follow the original works on the subject [6–11].

2.1 Three-dimensional supersymmetric theories

To begin, we review the field content of the 3d N = 4 multiplets. An N = 4 hypermultiplet

Q consists of a pair of N = 2 chiral superfields, Q and Q, in conjugate representations of

the gauge group. The N = 4 vector multiplet V consists of an N = 2 vector superfield V

and an N = 2 chiral superfield Φ. Recall as well that the N = 2 vector superfield contains

a real scalar field σ (the extra gauge field component in the dimensional reduction from

4d), so each N = 4 vector multiplet gives rise to a triplet of scalar fields.

Our notation for the component fields in general will be as follows. The Bose compo-

nents of V consist of the gauge field Aµ, the scalar σ mentioned above, and an auxiliary

field D. The gaugino λ is its Fermi component. Φ contains a complex scalar φ as well

as fermions ψ and an auxiliary field F . The scalars in Q, Q will be denoted by q, q, while

the fermions will be ψq, ψq and so forth. We work in (+ − −) signature and follow the

conventions for three dimensional SUSY of [22].

The simplest N = 4 theory is that of a free hypermultiplet Q = (Q, Q). In N = 2

notation,

LH(Q) =

∫d4θ(Q†Q+ Q†Q) = |∂µq|2 + |∂µq|2 + iψqγ

µ∂µψq + iψqγµ∂µψq . (2.1)

We always take the fermions to be Dirac, and ψ = ψ†γ0. This theory has a global U(1)

symmetry under which Q has charge +1 and Q has charge −1. Since we will be interested

in understanding the effects of x-dependent background fields (related to the insertion of

defects), let us introduce a background vector multiplet V = (V , Φ) for the U(1) symmetry

of this simple theory. This modifies the Lagrangian to

LH(Q, V) =

∫d4θ(Q†e2VQ+ Q†e−2V Q) +

∫d2θ√

2ΦQQ+ c.c. (2.2)

In components,

LH(Q, V) = |Dµq|2 + |Dµq|2 + iψq 6Dψq + iψq 6Dψq −(σ2 + 2|φ|2

)(|q|2 + |q|2

)−σ(ψqψq − ψqψq)−

√2(φψqψq + c.c.) +

√2(i¯λ(q†ψ − q†ψq) + c.c.

)+D

(|q|2 − |q|2

)+√

2(F qq + c.c.) (2.3)

where the covariant derivative

Dµϕi ≡ (∂µ + ieiAµ)ϕi (2.4)

for a field ϕi of charge ei. The background scalars σ and φ give real and complex masses,

respectively; we also note the possibility of background D- and F-terms, that will appear

in our analysis in later sections.

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JHEP11(2013)004

Next, let us consider an N = 4 U(1) gauge theory with vector-multiplet V and a

charged hypermultiplet Q. The Lagrangian is

L = LH(Q,V) + LV (V) (2.5)

where LH(Q,V) is given by (2.2) with the replacement V → V, and the kinetic terms for

the vector superfield are

LV (V) =1

2g2

∫d2θWαW

α +1

g2

∫d4θΦ†Φ (2.6)

=1

g2

[− 1

4F 2µν +

1

2(∂µσ)2 + |∂µφ|2 + iλ 6∂λ+ iψφ 6∂ψφ +

1

2D2 + |F |2

],

where Fµν = ∂µAν − ∂νAµ. The Lagrangian for more general N = 4 field contents can be

found in the obvious way. For instance, in the event that there are multiple hypermultiplet

flavors one simply adds an i index to the q and q fields with i = 1, · · · , Nf .

There is one further symmetry that will be useful to keep in mind. In three dimensions,

any abelian gauge field gives rise to a global U(1)J current via the relation

Jµ =1

2εµνρF

νρ. (2.7)

It is clear that the ‘charge’ J0 of this U(1)J symmetry is carried by configurations with

nonzero magnetic flux — i.e., vortices. This symmetry shifts the dual photon

Fµν = εµνρ∂ργ (2.8)

by a constant.1 This symmetry plays a central role in mirror symmetry, as we review

shortly. We can consider turning on a background vector superfield V for the global U(1)J ,

which then couples to V via a BF interaction,

LBF (V, V) =1

2π

(1

2εµνρAµFνρ + σD + σD + φF + φF + fermions + c.c.

). (2.9)

The background Aµ gives a chemical potential or magnetic flux source for the dynamical

gauge field, and σ is an FI term. The other contributions are supersymmetric generaliza-

tions of these. Note that LBF is the supersymmetric version of coupling the dual photon

to an external gauge field, L ⊃ JµAµ, with Jµ defined in (2.7).

The N = 2 gauge theory can be obtained from the N = 4 version by erasing the chiral

multiplet Φ. It is also useful to connect the two theories by RG flows. Starting from the

N = 4 theory, we can add a chiral superfield S and couple it supersymmetrically to φ via

the superpotential W = SΦ. This makes S and Φ massive, and in the IR we obtain the

N = 2 gauge theory. Similarly, starting from the N = 2 theory we can add Φ and perturb

by the superpotential W = QΦQ; this interaction is relevant and makes the theory flow to

a point with enhanced N = 4 supersymmetry.

1With this normalization, the dual photon has kinetic term L ⊃ 12g2

(∂µγ)2. Also, charge quantization

(see e.g. [23]) implies that it is compact with period γ → γ + g2.

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JHEP11(2013)004

Finally, since we will discuss theories at finite density and/or magnetic field that respect

some supersymmetry, we will need the supersymmetry variations. The variations for an

N = 2 supersymmetry (Dirac) spinor ε are as follows. For a chiral superfield Φ = (φ, ψ, F ),

δφ = εψ

δψ = (−i 6Dφ− σφ)ε+ Fε† (2.10)

δF = ε†(−i 6Dψ + σψ + iλφ) ,

where Dµ is the covariant derivative introduced before. For a vector superfield V =

(Aµ, σ, λ,D), the variations are

δAµ =i

2(εγµλ− λγµε)

δσ =i

2(ελ− λε)

δλ =

(1

2γµγνFµν− 6∂σ − iD

)ε (2.11)

δD =1

2(ε 6∂λ+ ∂µλγ

µε) .

2.2 Mirror symmetry for N = 4 theories

The mirror pair of theories we will be interested in is the pair given by the N = 4 abelian

gauge theory with one charged hyper Q on the one hand, and the free theory of an N = 4

hyper (which we denote by Q = (V+, V−)) on the other. We will refer to these as the

“electric” and “magnetic” theories, respectively. This is the simplest example of [6], and

mirror symmetry relating these theories can be proven along the lines of [11]. Mirror

symmetry states that both theories flow to the same infrared fixed point, so that their

partition functions with external sources become equal:

Zelectric[V] = Zmagnetic[V] . (2.12)

Here

Zelectric[V] =

∫DV DQ ei

∫d3x (LH(Q,V)+LV (V)+LBF (V,V)) (2.13)

and

Zmagnetic[V] =

∫DQ ei

∫d3xLH(Q,V). (2.14)

Let us discuss the implications of the duality in more detail. N = 4 theories enjoy

a global SU(2)L × SU(2)R R-symmetry. In the abelian gauge theory, SU(2)L acts on the

three scalars in the N = 4 gauge multiplet as a triplet, while leaving q, q invariant; SU(2)Racts on (q, q∗) as a doublet. In the free hypermultiplet theory, (v+, v

∗−) form a doublet of

SU(2)R. Mirror symmetry exchanges SU(2)L in the QED theory with SU(2)R in the free

hypermultiplet theory. It also maps the external sources V according to (2.12). In partic-

ular, the electric theory has a triplet of FI terms (σ, φ), which are spurions transforming

under SU(2)R; they are mapped to the real and complex masses in the magnetic theory,

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JHEP11(2013)004

which are spurions of the SU(2)L. In the mirror description, U(1)J acts simply as ±1 on

the V± chiral multiplets. In this sense, mirror symmetry acts as particle/vortex duality.

In order to see how the moduli spaces map, recall that the gauge theory in the pair

enjoys a scalar potential

V =1

2g2

(|q|2 − |q|2 +

σ

2π

)2

+ g2

∣∣∣∣√2qq +φ

2π

∣∣∣∣2 +(σ2 + 2|φ|2

)(|q|2 + |q|2

). (2.15)

The first two terms come from integrating out the auxiliary fields in the vector multiplet,

while the last terms come from the F-terms of the matter fields and the interaction with

σ can be understood as coming from the fourth component of the gauge field in 4d. In

N = 2 language, the F-terms arise from the superpotential W =√

2 QΦQ. The effect

of the background FI terms (σ, φ) is included for later applications. The theory has a

Coulomb branch of vacua parametrized by φ, σ and the dual photon γ defined in (2.8).

There is no Higgs branch with Nf = 1 hypermultiplet, the case we are focusing on. In

addition, quantum-mechanically the origin of the Coulomb branch is lifted; see e.g. (2.18)

below.

The Coulomb branch of N = 4 SQED maps to the moduli space spanned by v±.

Classically, one can identify2

v± ∼ exp

(± 2π

σ + iγ

g2

). (2.16)

On the Coulomb branch ~φ ≡ (σ, 1√2

Reφ, 1√2

Imφ) the gauge coupling function receives

one-loop corrections,1

g2L

=1

g2+

1

4π|~φ|(2.17)

while higher loop and nonperturbative corrections are absent for abelian N = 4 theo-

ries. The quantum-corrected moduli space is given by a sigma model with Taub-NUT

metric [6, 7],

Leff =1

2g2

(H(φ)(∂µ~φ)2 +H−1(φ)

(∂µγ +

1

2π~ω · ∂µ~φ

)2)

(2.18)

with

H(φ) = 1 +g2

4π|~φ|, ~∇× ω = ~∇H . (2.19)

The IR limit g2/|φ| → ∞ of this sigma model is the mirror free hypermultiplet theory.

This can be seen by redefining the fields (see e.g. [12])(v+

v∗−

)=

√|~φ|2πe2πiγ/g2

(cos θ2eiλ sin θ

2

), ~φ = |~φ|(cos θ, sin θ cosλ, sin θ sinλ) (2.20)

2This follows from the U(1)J charges of V±, the periodicity of the dual photon γ → γ + g2, and the fact

that the tree level kinetic term fixes the holomorphic coordinate to be σ + iγ.

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JHEP11(2013)004

which indeed yields the free Lagrangian

Leff = |∂µv+|2 + |∂µv−|2. (2.21)

A similar change of variables gives the fermions in the hypermultiplet. This mirror map

between the electric and magnetic theories will be quite useful later. It will allow us to

find the IR fate (in the dual variables) of solutions of the SQED theory, by mapping them

to exact solutions in the dual.

2.3 Mirror symmetry for N = 2 theories

The N = 2 mirror pair we will consider is a simple modification of the N = 4 example.

As we discussed in section 2.1, in the electric theory we can flow from N = 4 to N = 2

by adding a chiral superfield S with superpotential interaction W = SΦ. Both S and Φ

become massive, and in the IR one is left with the N = 2 U(1) gauge theory. This theory

has both a Coulomb branch parametrized by σ+2πiγ, and a Higgs branch where the meson

M = QQ gets an expectation value.

This flow is useful because it allows one to determine the mirror N = 2 theory. The

nontrivial Higgs branch of the electric theory implies that the magnetic theory should

contain an additional chiral superfield M , besides V±. Since the F-term for Φ in the

electric theory sets S = QQ and the mapping of the moduli space is Φ ∼ V+V−, the

superpotential deformation in the magnetic theory becomes

W = hMV+V− , (2.22)

where h is a coupling with dimensions of mass1/2. Its moduli space of vacua therefore

has three branches, depending on which of the (complex scalars in the) chiral multiplets is

non-vanishing. They meet at an interacting conformal field theory at the origin, which is

a supersymmetric generalization of the Wilson-Fisher fixed point.

At low energies, or equivalently close to the origin of the moduli space g2/|σ| � 1, the

quantum-corrected SQED coupling grows small according to (2.17). This means that the

radius of the dual photon (which is proportional to g2 in the UV) shrinks to zero at the

origin of moduli space. Quantum corrections therefore ‘split’ the common meeting locus of

the Higgs branch and Coulomb branch in the gauge theory into a junction between three

cones, as in figure 1. This agrees with the classical moduli space of vacua of the mirror

with superpotential (2.22).

Finally, the global symmetries of the electric and magnetic theories are given by

U(1)R U(1)J U(1)AQ 0 0 1

Q 0 0 1

M 0 0 2

V± 1 ±1 −1

(2.23)

Here U(1)R is the N = 2 R-symmetry, U(1)J is the topological symmetry ?F of the gauge

theory discussed before, and U(1)A is a global axial symmetry. The dual photon acquires

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JHEP11(2013)004

M

V V - +

Figure 1. The quantum moduli space of the N = 2 SQED theory near the origin agrees with the

classical moduli space of the supersymmetric Wilson-Fisher theory.

axial charge due to a one-loop BF term, while this effect is seen at tree level in the mirror

from (2.22).

Unlike the N = 4 mirror pair, here we have a strong/strong duality, valid at energy

scales much smaller than the relevant couplings of the electric and magnetic theory. While

neither side provides a weakly coupled description of the long distance physics, the duality

is still physically interesting, connecting a supersymmetric version of the Wilson-Fisher

fixed point to a theory with an emergent U(1) gauge field. The mapping between particle

and vortex excitations also plays an important role in understanding the dynamics in the

presence of defects, to which we turn next.

3 SUSY defects and mirror symmetry in N = 4 theories

Now we are ready to consider the addition of external electric or magnetic sources, which

amount to turning on finite density and/or magnetic fields. Our first goal is to deter-

mine whether these sources can preserve some supersymmetry. We find that it is possible

to have finite density or magnetic fields that preserve half of the supercharges. This is

an important step, because it allows us to construct and study general (possibly space-

dependent) configurations by superposing half BPS pointlike defects. In the second part

of our analysis, we will use mirror symmetry to understand the IR dynamics in the pres-

ence of supersymmetric defects. UV sources which interact with a strongly coupled theory

backreact on the field configuration in a way which is summarized by the mirror solution.

We also discuss how some of our results can be interpreted in terms of insertions of Wilson

and ’t Hooft line operators. For related work on such operators in the context of mirror

symmetry see [24, 25].

3.1 Adding electric charges to N = 4 SQED

We would like to add (static) external charges, with some charge-density ρ(x), to the theory.

We accomplish this by adding a source term for the U(1) gauge field

L ⊃ 1

2πρ(x)A0 . (3.1)

However, the charge density by itself breaks supersymmetry: the action is no longer in-

variant under the SUSY variations (2.11).

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Let us view the density as an expectation value for a background gauge field Aµ that

couples to the dynamical Aµ via a BF interaction L ⊃ εµνρAµF νρ. Because of the form of

this coupling, a charge density is obtained from a background magnetic field,

ρ(x) =1

2εijF

ij , (3.2)

a relation that will have important consequences for the mirror description at long dis-

tances. In order to preserve some supersymmetry, we need to turn on additional sources in

the background vector superfield V (which has Aµ as one of its components) and allow for

the supersymmetrization of the BF interaction, eq. (2.9). Specifically, we add the following

source terms to the N = 4 SQED theory:

Lsource =1

2π

(1

2A0 ε

ijFij(x) + σD(x)

), (3.3)

where Fij and D depend on space but not on time.

Now, let us imagine weakly gauging V; supersymmetry will be preserved if the variation

of the gaugino λ vanishes,

δλ =

(1

2γiγjFij(x)− iD(x)

)ε = 0 . (3.4)

Recalling that the gamma matrices in 2 + 1 dimensions satisfy (in our conventions)

γµγν = gµν1− iεµνργρ , (3.5)

half of the supersymmetries are preserved, (1± γ0)ε = 0, as long as

1

2εijF

ij(x) = ± D(x) . (3.6)

Therefore, it is possible to have supersymmetry at finite density as long as we add the

extra source D determined by (3.6).

Another way of proving (3.6) — which does not weakly gauge V — is to require that

the solutions of the equations of motion in the presence of sources preserve supersymmetry.

Working for simplicity at weak coupling and ignoring the interactions with matter fields,3

the equations of motion in the presence of the sources are

1

g2∂2i A0 =

ρ(x)

2π,

1

g2∂2i σ(x) = −D(x)

2π. (3.7)

The index i runs over the spatial directions. Now, supersymmetry requires that the SUSY

variations of all fermions should vanish on the background bosonic field configuration. The

gaugino variation δλ = 0 imposes, from (2.11),

(∂iA0γ0 − ∂iσ)ε = 0 , (3.8)

3This is a good approximation far along the Coulomb branch.

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JHEP11(2013)004

where we have set D = 0. In order to preserve half of the supercharges, we need to impose,

up to a constant,

A0 = ±σ , (3.9)

with the unbroken supercharges being (1 ∓ γ0)ε = 0. From (3.9) together with (3.7),

this amounts to a condition on the sources D = ∓ρ, which gives indeed (3.6). This

approach is equivalent to the requirement that the Lagrangian is invariant under the SUSY

transformations eq. (2.11).

In summary, an external charge distribution added to the path integral via an inser-

tion of

Wρ = exp

(− i

2π

∫dtd2xρ(x)(A0 ± σ)

)(3.10)

can preserve half of the supersymmetry. In the special case where ρ(x) = δ2(x), this is

simply the familiar 1/2 BPS Wilson line of supersymmetric gauge theories.

3.1.1 Classical solutions and field of a point charge

We can present solutions to the equations of motion in the UV (where g → 0) for rather

general sources. The dynamics at long distances is strongly coupled and will be analyzed

below using the mirror dual description.

Let us switch for convenience to complex coordinates

z =x1 + ix2√

2, ∂z =

∂1 − i∂2√2

, Az =A1 − iA2√

2(3.11)

with the obvious definitions for complex conjugates. We recall that in two dimensions

∂∂ log(zz) = 2πδ2(z, z) . (3.12)

Equipped with this Green’s function, we can immediately write down the UV solution

generated by a given charge distribution:

A0(z, z) = ±σ(z, z) =g2

8π2

∫d2u ρ(u, u) log |u− z|2. (3.13)

One is free to add a homogeneous solution to (3.7), which corresponds to shifting

A0 → A0 + f(z) + f(z) (3.14)

(where f must be the complex conjugate of f to keep the gauge field real). Boundary con-

ditions at infinity can determine the homogeneous solution. For instance, for a spherically

symmetric distribution of external charges, we would wish to find a spherically symmetric

solution and set f(z) = 0.

The dual photon also has interesting behavior. From its definition (2.8), it follows that

∂zγ = i∂zA0, ∂zγ = −i∂zA0. The solution to these equations is

γ(z, z) = ig2

8π2

∫d2u ρ(u, u) log

(u− zu− z

). (3.15)

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JHEP11(2013)004

On the Coulomb branch, the σ scalar in the gauge supermultiplet combines with γ to form

a complex scalar, whose solution is then

σ + iγ =g2

8π2

∫d2u ρ(u, u)

[± log |u− z|2 − log

u− zu− z

]. (3.16)

Therefore supersymmetric solutions have either a holomorphic or antiholomorphic profile

for σ + iγ.

As the simplest example, consider a point-like external charge,

ρ(z, z) = 2πq0 δ2(z, z) . (3.17)

The gauge field and σ take the form

A0(z, z) = ±σ(z, z) = − 1

4πg2q0 log

|z|2

r20

, (3.18)

where the constant r0, added for dimensional reasons, is related to the Coulomb branch

expectation value. The dual photon becomes

γ = −g2q0θ

2π(3.19)

and θ is the standard angular variable on the complex plane, z = reiθ. Note that when going

around an electron of unit electric charge the dual photon has a monodromy γ → γ + g2,

which equals its periodicity.

Of course, the profile of the electric potential for external charges in the non-interacting

limit is well known. However, we will next consider the long distance behavior, and find

a much more interesting solution. Surprisingly, strong screening effects from the chiral

multiplets (Q, Q) turn the logarithmic running into a constant. Also, the linear superpo-

sition (3.13) will flow to a product of monomials, each of which can be interpreted as a

localized vortex for the topological U(1)J .

3.2 Global vortices in the mirror configuration

As one flows to the IR in the N = 4 SQED theory, g → ∞ and the magnetic description

in terms of the free hypermultiplet V± is more appropriate. We will use the mirror map,

reviewed in section 2.2, to find the long distance dynamics of the supersymmetric finite

density configuration. Let us first directly study this magnetic theory and then map it to

the electric variables.

According to (2.12), an external charge density in the electric theory corresponds to a

U(1)J background magnetic field in the mirror dual, while the electric source term D is a

background D-term. As before, the sources can have arbitrary dependence on space, but

need to be static. The Lagrangian for the bosonic fields takes the form

L = |Dµv+|2 + |Dµv−|2 + D(x)(|v+|2 − |v−|2

), (3.20)

where Dµv± = (∂µ ± iAµ)v±.

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We wish to find supersymmetric configurations with a non-zero static magnetic field

B = F12, or in complex coordinates

∂zAz − ∂zAz = iB(z, z) . (3.21)

Weakly gauging the global symmetry and setting to zero the SUSY variation of the gaugino

δλ = 0 gives, as in (3.6),4

B(x) = ±D(x) , (1± γ0)ε = 0 . (3.22)

However, since the magnetic theory is free, it is more instructive to follow a different route,

which will also yield the first order equations that need to be satisfied by v±.

Unbroken supersymmetry requires5

δψ± = −i(γzDzv± + γ zDzv±)ε = 0 . (3.23)

Half of the supersymmetries are preserved if

γzε = 0 , Dzv± = 0 (3.24)

or

γ zε = 0 , Dzv± = 0 . (3.25)

On the other hand, the equations of motion following from (3.20) are

(DzDz +DzDz)v± ± D(z, z)v± = 0 . (3.26)

Let us choose the SUSY case (3.24), for which Dzv± = 0. Plugging into (3.26), and since

in the presence of the external B field,

[Dz, Dz]v± = ∓B(z, z)v± , (3.27)

we find that the equation of motion can be solved if

B(z, z) = −D(z, z) . (3.28)

In the other case (3.25) one instead requires B = D. This reproduces (3.22).

We conclude that, in the presence of an external magnetic field (with arbitrary space

dependence), the free hypermultiplet theory admits solutions that preserve half of the

supercharges if an external D-term (3.22) is also turned on.

4This approach was used in e.g. [20] in the context of supersymmetric Landau levels.5According to our conventions (3.11), γz = 1√

2(γ1 + iγ2), Dz = ∂z + ieAz, Dz = ∂z + ieAz, etc.

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3.2.1 Classical solutions and external vortices

Let us now study the supersymmetric classical solutions in the presence of an external

magnetic field and D-term D(x) = ±B(x).

Unlike the electric theory where only the field strength Fµν appears, here the hyper-

multiplet couples to the background potential Aµ, so we need to choose a gauge. Given an

external magnetic field B, a convenient choice that solves (3.21) is

Az = − i

4π∂z

∫d2u B(u, u) log |u−z|2, Az =

i

4π∂z

∫d2u B(u, u) log |u−z|2, (3.29)

which uses the Green’s function on the plane.

With B = −D, the supersymmetric configurations satisfy Dzv± = 0, (3.24). Plugging

in the background gauge field (3.29), it is straightforward to integrate this equation to

obtain

v±(z, z) = f±(z) exp

(± 1

4π

∫d2u B(u, u) log |u− z|2

), (3.30)

with f± arbitrary holomorphic functions. Similarly, with B = D, the solutions to

Dzv± = 0 are

v±(z, z) = f±(z) exp

(∓ 1

4π

∫d2u B(u, u) log |u− z|2

). (3.31)

These arbitrary (anti)holomorphic prefactors correspond to the gauge freedom noted above,

while a constant prefactor moves the configuration along the Higgs branch of the hypermul-

tiplet theory. In order to determine the physically correct choice of f±, it is instructive to

first discuss pointlike magnetic flux insertions, from which we can then construct a general

B by superposition.

Choosing a delta-function localized magnetic flux

B(z, z) = 2πq0δ2(z, z) , (3.32)

(3.29) gives simply A = q0dθ. Any loop encircling the insertion at r = 0 will have a fixed

holonomy ei∮A = e2πiq0 , which is the definition of a global vortex. Every charged field

picks an Aharonov-Bohm phase given by the magnetic flux times its charge. This fixes the

previous arbitrary prefactor to f± ∝ z∓q0 , obtaining the solution

v± = v±0

(z

z

)±q0/2= v±0 e

∓iq0θ (3.33)

for Dzv± = 0, and the complex conjugate of (3.33) for the SUSY case Dzv± = 0. The

constant v±0 parametrizes the Higgs branch position.

Eqs. (3.32) and (3.33) represent an external (nondynamical) pointlike vortex. Let’s

compare this to an Abrikosov vortex. Recall that the abelian Higgs model in 2 + 1 di-

mensions admits dynamical vortices where the behavior of the scalar field at infinity is

φ → veiNθ, with v the minimum of the potential and N the winding number. At long

distances, A ∼ Ndθ and the magnetic flux is proportional to N . The vortex has a core

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JHEP11(2013)004

where the scalar and gauge field vanish. In the present case, the magnetic flux is exter-

nal and delta-function localized. The scalars in the hypermultiplet have a Higgs branch

parametrized by v±0 above, and the effect of the nondynamical vortex is to introduce a

nonzero winding, (3.33), which is singular at the location of the pointlike core.

Returning now to the general case, the previous discussion fixes f± to give a mon-

odromy determined by the total magnetic flux,

f±(z) = v±0 exp

(∓ 1

2π

∫d2u B(u, u) log(u− z)

). (3.34)

In conclusion, the general supersymmetric configurations that reproduce the correct mon-

odromies around the vortex insertions are given by

B = D , (1 + γ0)ε = 0 , Dzv± = 0 , v± = v0 exp

(± 1

4π

∫d2u B(u, u) log

u−zu−z

),

(3.35)

and

B = −D , (1− γ0)ε = 0 , Dzv± = 0 , v± = v0 exp

(∓ 1

4π

∫d2u B(u, u) log

u−zu−z

).

(3.36)

3.2.2 Mapping to the electric variables

Finally, consider the map from the electric to the magnetic theory given by (2.20), a result

which is exact in the regime |~φ|/g2 � 1. The mapping of external sources is ρ(x) = B, and

D is the same on both sides. This means that an external electron of the SQED theory maps

to a global U(1)J vortex in the magnetic theory. For a pointlike charge ρ = 2πq0δ2(z, z),

the electric theory dual photon has monodromy g2q0. In the magnetic theory, this effect

appears as the Aharonov-Bohm phase of the charged v±. Therefore, the phases of the

map (2.20) agree, as they should.6 The duality between a Wilson line insertion in the

SQED theory and a vortex in the magnetic dual is part of the particle/vortex duality in

mirror symmetry and has been made more precise in e.g. [24, 25].

On the other hand, σ has a much more interesting behavior at different scales. For a

point charge, σ diverges logarithmically in the electric theory. However, the behavior at

long distances is given by (2.20),

|σ| = 2π|v±|2, (3.37)

where for simplicity we have set the remaining Coulomb branch coordinates to zero. Re-

calling (3.33), we see that |σ| flows to a constant in the IR. The logarithmic behavior has

been screened by the strong dynamics of (Q, Q) in the electric theory, in a way that is

captured by the classical solution of the magnetic dual theory.

The mapping with multiple local sources (or a more general spatially dependent dis-

tribution) is also very interesting. In the electric theory, for a solution with ρ(x) =

6In both theories, there is a conserved U(1) symmetry from a combination of spatial rotations and a

U(1)J transformation. With the f± found before, the same linear combination is preserved on both sides

of the duality.

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JHEP11(2013)004

ρ1(x) + ρ2(x), in the UV g → 0 limit, the solutions are additive — one simply uses the

Green’s function to add up the contributions due to the two localized sources.

By the time one flows to the IR, we see that the superposition of UV sources has

created considerably more complicated effects. ρ(x) maps directly to B(x) under mirror

symmetry. The solution created by a superposition B(x) = B1(x) + B2(x) (with each Biresulting from the localized source in the electric theory) is the exponential (3.35), exhibit-

ing additivity in the exponent and a very complicated interaction between the sources!

The field configurations (3.35) and (3.36) capture the complicated process (due to strong

dynamics) by which the field configuration sourced by external charges in N = 4 SQED

changes as one flows to the IR. We will use this in section 5 to solve for the behavior of

defects in N = 4 SQED as one flows to strong coupling.

3.3 Adding magnetic charges to N = 4 SQED

Now we consider the reversed situation, where we turn on a source for the U(1) magnetic

field of the SQED theory. In the mirror dual, this corresponds to a chemical potential for

U(1)J . We will first study the conditions under which this can be done in a supersymmetric

fashion before analyzing the solutions in the presence of spatially dependent sources. An

important difference with the previous situation is that now the theory will admit BPS

configurations of finite central charge, and this will allow us to understand various aspects

of the correspondence between particles and vortices across the duality.

As before, the sources are part of the N = 4 background vector multiplet V, which

appears in the terms (2.9). A source for ∂iAj is given by a nonzero A0(x). Following the

steps of the previous sections, we weakly gauge V and impose δλ = 0, eq. (2.11). This

shows that the other source that needs to be turned on in order to preserve SUSY is the

background scalar σ, which plays the role of an FI term for the U(1) gauge theory. We

study the SQED theory in the presence of

Lsource =1

2π

(A0(x) εij∂iAj + σ(x)D

). (3.38)

The solution to (γ0∂iA0 − ∂iσ)ε = 0 is

A0(x) = ±σ(x) , (1∓ γ0)ε = 0 . (3.39)

At this stage, a somewhat subtle point needs to be addressed. Eq. (2.9) defined the BF

interaction to be of the form εµνρAµFνρ, namely a coupling of the topological current ?F to

an external gauge field. Furthermore, this form of interaction is explicitly gauge invariant.

We could have chosen instead an interaction εµνρAµFνρ, differing from the previous one

by a boundary term. The equations of motion are not modified, and the boundary term

vanishes if the external source falls off fast enough at infinity. However, we will be interested

in the possibility of constant FI terms; then A0 and the boundary term do not vanish at

infinity. The correct form in such cases is (3.38). This has important consequences for the

vacuum structure of the theory, as we discuss shortly.

Let us now turn to the field configurations; we will see that (3.39) also follows from

requiring that the solutions to the equations of motion preserve SUSY. We will consider first

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JHEP11(2013)004

the theory at the origin of the Coulomb branch. In the presence of the background (3.38),

the dynamical magnetic field B = F12 and D-term will be turned on. The gaugino variation

δλ = 0 is satisfied for1

2εijF

ij = ±D , (1± γ0)ε = 0 , (3.40)

where the auxiliary field is

D = −g2

(|q|2 − |q|2 +

σ(x)

2π

). (3.41)

Furthermore, the conditions δψq = δψq = 0 that preserve half of the supercharges are

Dzq = Dz q = 0 , γzε = 0 (3.42)

or

Dzq = Dz q = 0 , γ zε = 0 . (3.43)

Next, compare these conditions with the classical equations of motion. The (q, q)

equations are solved for (3.42) if B = −D, while the other case (3.43) needs B = D. These

are consistent with the gauge field equation only if A0 = ±σ, agreeing with what we found

in (3.39). To summarize, the SUSY conditions at the origin of the Coulomb branch are

A0(x) = −σ(x) , B = D , Dzq = Dz q = 0

A0(x) = σ(x) , B = −D , Dzq = Dz q = 0 . (3.44)

The F-term for Φ requires qq = 0, so one of the scalars has to vanish. The solutions (if

they exist) preserve half of the supercharges. These would be generalizations of the BPS

vortices to the case of an x-dependent FI term and source A0. It would be interesting to

determine whether such solutions exist, and study their dynamics.

In the case where the FI term σ is a constant, the nontrivial BPS solutions to these

equations are well known. They are the familiar vortices of the abelian Higgs model, studied

in detail in supersymmetric theories in e.g. [26, 27]. In a background with∫d2x B = k

(i.e. k units of magnetic flux), these vortex solutions have a moduli space Mk which is k

(complex) dimensional. One can think of it as being spanned by the positions of the k

vortices in the plane, and having the asymptotic structure Ck/Sk when the vortices are

well separated.

There is, however, an important difference with the SUSY Higgs model: the

sources (3.38) include, besides the FI term, a coupling A0B to the magnetic field. For

constant A0 this term is a total derivative and so the equations of motion are not modi-

fied. However, taking into account the SUSY conditions, it gives a negative contribution

to the total energy. Due to this effect, the Coulomb branch is not lifted. Indeed, one can

show that

q = q = 0 , B = ±g2 σ

2π, (3.45)

solve the SUSY variations and equations of motion. This vacuum allows for Coulomb

branch expectation values, along which the vortices are lifted. This should be constrasted

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JHEP11(2013)004

with the theory with just an FI term and no A0, for which there is no Coulomb branch.7

Therefore, our supersymmetric extension of the FI term admits both vortex solutions and

a nontrivial Coulomb branch, without having to turn off the FI term. This will provide

a more precise mapping between vortices and particles in the mirror dual, to which we

turn next.

3.4 Finite chemical potential in the mirror dual

The magnetic dual involves turning on a chemical potential for U(1)J , as well as a real

mass σ for the chiral multiplets V±. The Lagrangian is

L = |Dµv±|2 + iψ± 6Dψ± − |σ|2(|v+|2 + |v−|2

)− σ(ψ+ψ+ − ψ−ψ−) (3.46)

The supersymmetry variations of the fermions are now (for constant v±)

δψ± = −(i 6Dv± ± σv±)ε = ± v±(γ0A0 − σ)ε (3.47)

Unbroken supersymmetry requires, unsurprisingly, that A0 = ±σ, with half of the super-

symmetry being preserved when this is the case. With this relation between the sources,

the positive scalar mass squared from σ2 exactly cancels the negative contribution from

the chemical potential, leaving

L = |∂µv±|2 + iψ± 6∂ψ± ∓ iA0(v∗±∂0v± − v±∂0v∗±)∓ A0 ψ±(1 + αγ0)ψ± , (3.48)

with α = ± the relative sign between A0 and σ.

Unlike the case with only a chemical potential (which leads to tachyonic scalars), here

there is a Higgs branch moduli space where v+ and v− have arbitrary constant values.

Therefore the vacuum is a superfluid where the order parameter for symmetry breaking is

not fixed. This is the mirror dual of the Coulomb branch that appears for (3.45).

Let us discuss in more detail the particle excitations in the presence of a constant

background σ field and the chemical potential. Since (1±γ0)/2 is a projector on the spinor

indices, the background gives half of the fermion components in each ψ± a mass 2A0,

while the other half are massless. Intuitively, the real mass σ in the absence of a chemical

potential would give a mass to all of the particles. The presence of the chemical potential

decreases the mass of the antiparticles to zero, and increases the mass of the particles (or

the reversed, depending on the sign of A0). Similarly, the scalar equation of motion in this

class of backgrounds becomes

�v± = ±2iA0 ∂0v± , (3.49)

which admits plane-wave solutions of frequency ω = A0 ±√A2

0 + k2 for v+, and a similar

solution for v− with A0 → −A0. The bosons and fermions have degenerate masses, as

expected from supersymmetry. The massless root describes an infinitesimal fluctuation

along the Higgs branch, while the massive root, with ω = 2A0 at k = 0, corresponds to a

massive BPS particle.

7Finite energy requires a Higgs branch vacuum |q|2 − |q|2 + σ/(2π)→ 0 at infinity.

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JHEP11(2013)004

We are now in a position to describe the mirror configuration to k BPS vortices.

Because the U(1)J current is ?F , a magnetic field for the dynamical gauge field maps to the

presence of U(1)J charges in the free hypermultiplet description. The mirror configuration

is then precisely k of the massive BPS particles discussed above. The moduli space of

the k positions of these particles maps to the moduli space of k abelian vortices in the

SQED description (though the detailed geometry of the moduli space is expected to be

different). Furthermore, the superfluid phase where the U(1)J is broken along the Higgs

branch (v+, v−) maps to the Coulomb branch, where the vortices in the electric theory

are lifted. We can interpolate smoothly between the isolated vacua with vortices and the

Coulomb branch without changing the BPS spectrum or external sources, a phenomenon

which is manifest classically in the free hypermultiplet dual. This gives a very explicit

realization of the particle/vortex duality.

4 SUSY defects and mirror symmetry in N = 2 theories

We now discuss the dynamics of defects in the N = 2 theories of section 2.3. In three

dimensions, this is the smallest amount of supersymmetry for which there exist SUSY

defects; the reason is that our mechanism needs a vector multiplet (either gauge or global)

that contains both a scalar and a gauge field.

4.1 Semiclassical solutions

At the semiclassical level, the story with 3d N = 2 supersymmetry is quite similar to the

N = 4 theory. The main difference is that the electric theory has no analog of the chiral

superfield Φ (and hence W =√

2QΦQ is absent), while the magnetic theory has a new

singlet superfield M with a superpotential coupling W = hMV+V−. As a consequence,

there are new branches in the moduli space of vacua — the Higgs branch with q, q 6= 0 in

the N = 2 SQED theory, and the (mirror) M-branch in the N = 2 Wilson-Fisher theory.

The semiclassical story for the SUSY defects then carries over as follows. Consider first

adding electric charges to the SQED theory. The classical solutions in the UV are the same

as in section 3.1; the field Φ didn’t play an important role for the N = 4 defects (which

introduced sources for A0 and σ), and is absent in the N = 2 case. The magnetic dual

has nonzero U(1)J magnetic field and D-term backgrounds, and the solutions are global

vortices of the type described in section 3.2. However, unlike the N = 4 case, now the

F-term for M forces either v+ or v− to vanish. This maps to the statement that the field

Φ is absent from the electric theory. Furthermore, M = 0 along the v+ or v− branches.

The mapping of the M -branch is more nontrivial. Note that in the magnetic theory we

can set v+ = v− = 0 because the backgrounds only multiply quadratic functions of v±. This

should be contrasted with the electric theory, where σ and the dual photon acquire spatial

dependence and cannot vanish for generic sources. At the origin of the v-branches, the M -

branch opens up. What happens in the electric theory is that q and q obtain positive masses

proportional to σ2, but these are exactly cancelled by the negative contribution from A20.8

8This is similar to what we found in (3.48), but for a dynamical gauge field instead of a background.

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JHEP11(2013)004

Therefore, even though σ is nonzero, (q, q) can have arbitrary expectation values subject

to the D-flatness condition |q| = |q|, in agreement with the dimension of the M -branch.

Next, let us briefly discuss SQED theory in the presence of magnetic sources, the

N = 2 analog of section 3.3. Now qq = 0 is no longer an F-term condition, so q and q can

be varied keeping |q|2 − |q|2 fixed. This maps to the modulus M of the mirror dual. Also,

as explained before, the absence of the field φ in the electric theory maps to the F-term

v+v− = 0 in the dual. Taking these differences into account, the classical solutions are the

same as in the N = 4 case.

4.2 Comments on quantum dynamics

The quantum version of the N = 2 case is much richer than the N = 4 one, with both

electric and magnetic theories flowing to strong coupling in the IR. Both become equivalent

for energies E � g2 and E � |h|2, where g is the SQED gauge coupling and h is the

superpotential coupling (2.22). The Kahler potential in N = 2 theories is not protected,

so physical couplings and correlation functions receive large quantum corrections on both

sides.

The external electric or magnetic sources require Aµ and σ in the electric theory to be

nonzero, and similarly v± are turned on in the magnetic dual. These expectation values

need to be much smaller than the respective relevant couplings in order for the theories to

be dual. This is the regime of strong coupling, and the semiclassical solutions discussed

before will receive important quantum corrections. With the amount of supersymmetry

preserved by the defects (two supercharges), the expectation values of fields in the presence

of sources cannot be obtained analytically. In particular, the physical gauge coupling, which

determines the Coulomb branch metric, receives higher loop corrections in N = 2 theories,

in contrast with (2.17).

In some respects, this situation is similar to the nonsupersymmetric duality between

the U(1) abelian Higgs model and the XY model of [1, 2], where both sides are strongly

coupled. Some of the possible IR phases at nonzero chemical potential were discussed

in [28]. There are, however, important differences between the two systems. In the N = 2

duality discussed here there are flat directions that are protected by supersymmetry, while

these are absent in the nonsupersymmetric version. For example, in the magnetic theory

doped with U(1)J chemical potential in a supersymmetric fashion, we obtained a superfluid

phase with arbitrary expectation values 〈v±〉. This feature is expected to survive even at

strong coupling, at least if the external sources are localized within a finite region in

space. Another important distinction with the nonsupersymmetric case is the existence of

BPS excitations in the presence of external electric and magnetic sources. Because of the

additional control from supersymmetry it may be possible to analyze to some extent the

IR dynamics of doped N = 2 theories, a question which we hope to return to in the future.

5 Lessons for impurity physics

Mirror symmetry can be a powerful tool for elucidating the physics of impurities, which

is of significant interest in condensed matter physics (see e.g. [5, 29, 30]). This is clearest

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Figure 2. The solution for a single electrically charged defect in the UV limit of SQED.

in the duality between N = 4 QED and the free N = 4 hypermultiplet. For instance,

we may start with an arbitrary array of impurities in the electric theory (represented by

a configuration of localized sources in ρ(x)); map the sources to the magnetic theory via

the mirror map B = ρ; and then find the magnetic solution corresponding to the sources

using (3.35).

The physics of the IR solution in the electric theory can then be found by applying

the map between variables

v± =

√|σ|2π

e±2πiγ/g2(5.1)

(appropriate for solutions with θ = 0 in section 2.2). It is notable that the resulting solution

of the electric theory, applicable in the IR, is dramatically different from the naive solution

one would obtain using the Green’s functions of the classical UV theory to superpose effects

of the defect sources. We illustrate this with several examples now.

We start with a depiction of the basic electric defect in figure 2. This is the solution

from section 3.1 with

A0 = ±σ = − g2

4πq0 log |z|2. (5.2)

This isolated defect is mapped in the magnetic theory to v± = v±0 e±iq0θ. A plot of the

single vortex appears in figure 3. The constant coefficient v±0 determines which point on

the moduli space of vacua one approaches asymptotically. More precisely, for an external

vortex/anti-vortex pair, the winding at infinity vanishes, and one approaches some well

defined point in moduli space both in the electric and magnetic theories.

Mapping the single external vortex back to the IR electric theory using (5.1), one finds

an extremely simple dressed solution, σ ∼ const, γ ∼ arg(z). The dramatic change from

log growth to constant behavior visible in the dressed solution is an effect of screening of

the defect charge by the vacuum polarization of the strongly coupled SQED theory.

We can use this same technique to find the solution for an arbitrary array of δ-function

localized defects, such as a defect lattice. With localized sources at positions zi in the spatial

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JHEP11(2013)004

Figure 3. The backreaction of a single ‘external vortex’ in the magnetic theory.

Figure 4. A 3× 3 lattice of external vortices.

plane, the UV electric theory solution is

A0(z, z) = ±σ(z, z) = − g2

4π

∑i

qi log |z − zi|2, (5.3)

while the solution in the magnetic theory is

v± = v±0∏i

(z − ziz − zi

)±qi/2. (5.4)

A plot for a small 3× 3 periodic lattice in the magnetic theory appears in figure 4.

In the IR electric theory, the solution (5.4) becomes

|σ| = 2π|v±0 |2, γ = i

g2

4π

∑i

qi logz − ziz − zi

. (5.5)

There are basically several small cores around which there is a winding of the dual photon.

The ease with which one can find the analytical solutions for multiple-defect config-

urations, and analyze small fluctuations around the defect solutions, makes this a very

promising system for investigating defect effects on linear response and transport phenom-

ena in a strongly-coupled quantum gauge theory.

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JHEP11(2013)004

6 Non-supersymmetric deformations and ground-state entropy

It is easy to obtain controlled results for (slightly) non-supersymmetric theories by deform-

ing any of the previous results. For instance, when a configuration is supersymmetric if

ρ(x) = D(x) (sourcing A0 and σ in the N = 4 QED theory), deforming slightly to

ρ(x) = D(x) + δD , δD � D (6.1)

allows a controlled expansion about the more precisely calculable supersymmetric results.

Here, we use this philosophy to provide a simple example of a vexing phenomenon

which has shown up in the study of finite density systems in AdS/CFT. In that context,

to turn on a finite density of a global charge in the conformal field theory, one is instructed

to study a charged black brane geometry in the bulk AdS space-time. The simplest such

systems — charged black branes which arise in Einstein-Maxwell theory — give rise to a

puzzle. The extremal black brane (which corresponds to the ground state of the doped

field theory) has a non-vanishing entropy at zero temperature. The geometry of its near-

horizon region is AdS2×R2, and the fact that the horizon is extended implies an (extensive)

ground-state degeneracy.

This seems like a surprising result, as a doped, strongly interacting quantum field

theory would not in general be expected to have ground state entropy. (For instance, this

entropy violates ‘Nernst’s theorem’.) Discussions of this phenomenon can be found in [18–

20] and references therein. Here, we point out that our simple mirror pairs provide an

example of this phenomenon where the field theories involved are explicitly known (and

extremely simple).

Consider the N = 4 SQED theory in the presence of a constant density ρ of external

electric charges. This can be viewed as a limit of the case discussed in section 3.1 where we

have a lattice of electric charges and the theory is being analyzed at distances much larger

than the lattice spacing. The classical electric potential grows with distance, A0 = ±σ =ρg2

8π r2. The long distance physics is described by the mirror dual in in section 3.2 with a

constant magnetic field and D-term for the global U(1)J . This is a supersymmetric version

of the Landau level problem, where, because of the cancellation between the magnetic field

and D-term sources, both the scalars and the fermions have Landau level wavefunctions

with degenerate masses.9 The lowest Landau level has vanishing energy and preserves half

of the supersymmetries. Recalling the discussion around (3.24), the bosonic and fermionic

wavefunctions satisfy the first order equations Dzv± = 0 and Dzψ± = 0, or their complex

conjugates depending on the preserved supersymmetry.

Instead, we choose to break supersymmetry by turning on the B-field but leaving

D = 0. The condition for mirror symmetry to still be an approximate symmetry is that B

be small in units of the gauge coupling, an analogue of the condition (6.1) that prevents the

breaking of supersymmetry from leading to very large corrections to our statements. Now

in the theory with D = 0 but B 6= 0, the background magnetic field gaps the scalars in the

free hypermultiplet: they live in Landau levels, but with a positive zero-point energy for

the lowest Landau level (arising from the appropriate harmonic oscillator wavefunction).

9This system was also studied by [20].

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JHEP11(2013)004

0 1 2 3 4 5 6 7 8 9

D3 x x x x

2 NS5 x x x x x x

D5 x x x x x x

Figure 5. The brane configuration engineering the 3d N = 4 QED theory on a stack of D3-branes

(stretching between NS5 branes in the x6 direction).

The fermions, on the other hand, still have gapless modes. Their Hamiltonian is

H = ψ(γzDz + γ zDz)ψ . (6.2)

The Hamiltonian vanishes for γ zψ = 0 and Dzψ = 0, which comprise the lowest Landau

level for the fermion. An electron of charge e in a magnetic field B has an orbit of size

πr2 ∼ 1

eB(6.3)

which arises just by thinking about orbits of charged particles in a magnetic field. This

means that in a plane of area A, one can fit of order AB electrons in the lowest Landau

level (for a nice discussion in the context of the quantum Hall effect, see [31]).

We conclude that a system can develop a ground state degeneracy due to strong

dynamics, not seen in terms of the UV description. In a 2 + 1-dimensional theory, all

the spatial directions are threaded by a magnetic field, and the density of fermion zero

modes gives rise to a ground state entropy, precisely along the lines envisioned in [18–20].

The merit of this example is that the strongly coupled system is a well-known field theory

— the supersymmetric analogue of 2+1 dimensional quantum electrodynamics.10

7 D-brane picture

There are simple D-brane constructions of the various 3d supersymmetric field theories

we’ve studied [7, 8, 21], including the doping with external charges. This section reviews the

D-brane realization of theN = 4 andN = 2 theories and realizes the SUSY defects in terms

of semi-infinite F1 strings and D1 branes. This gives a geometric way of understanding

some of previous discussion.

7.1 Engineering the field theories

The 3d N = 4 QED theory with a single flavor can be constructed by the type IIB string

theory brane configuration depicted in figure 5.

The D5-brane location coincides with the D3-branes in the 345 directions, with the

D3-D5 strings giving rise to the massless hypermultiplet Q. The Coulomb branch of vacua

is parametrized by the D3 location along the NS5s in the 345 directions; this geometrizes

σ and Φ, but the dual photon is not geometrized by the brane construction.

10This system presents a counterexample to the well known Inverse Ninja Rule, in that the small N gauge

theory is more tractable than its large N counterparts. We thank E. Lim for discussions of this point.

– 23 –

JHEP11(2013)004

0 1 2 3 4 5 6 7 8 9

D3 x x x x

2 D5 x x x x x x

NS5 x x x x x x

Figure 6. The S-dual brane configuration to figure 5, giving the theory of a free hypermultiplet.

0 1 2 3 4 5 6 7 8 9

D3 x x x x

2 NS5 x x x x x x

D5 x x x x x x

F1 x x

Figure 7. The brane configuration for a delta function electric source in N = 4 QED, as described

in section 3.1.

The mirror brane configuration is obtained by the S-duality transformation of type

IIB string theory. This maps D3 branes to themselves, but takes NS5 branes to D5 branes

and vice-versa, as in figure 6.

As the boundary conditions freeze the D3 location in the 345 directions when it is

suspended between D5s, this gives rise to a field theory with no Coulomb branch. By

SUSY, one can infer (or compute directly) that there are no N = 4 vector multiplets in

this dual theory. However, the ‘half’ D3s can split their locations along the NS5 brane.

This gives rise to a Higgs branch of vacua, as the halves split to different locations in the

345 directions. (Again, one direction in the moduli space of vacua is not geometrized.)

The 3d N = 2 configurations are obtained in a similar manner. Rotating one of the

NS5 branes into an NS5’ brane (which wraps the 89 directions instead of the 45 directions)

breaks the SUSY in a suitable way. The Coulomb branch is partially lifted. The D3 may

no longer slide in the 45 directions while still ending on the NS5 and NS5’ branes. However,

the σ modulus still exists (from translations of the D3 along the 3 direction). In addition,

there is a new Higgs branch of vacua. Geometrically, this corresponds to moving a D3

segment between the D5 and NS5’ along the 89 directions. In the magnetic dual, one of

the D5s in figure 6 is rotated into a D5’ along the 89.

7.2 Including the background sources

The basic reason that we can easily describe our backgrounds in terms of brane sources is

that the delta function sources (which can give arbitrary sources by use of the appropriate

Green’s function) are geometrized nicely by string theory. For instance, in the N = 4 QED

theory, the basic charge which sources A0 and σ is in fact the fundamental string ending

on the D3-brane, as in figure 7 below. This result is of course well known in the literature

on Wilson loops in string theory.

The fact that the sources couple to A0 and σ is here a simple consequence of the

F1 worldvolume being extended along the 0 and 3 directions (the later corresponds to

the σ field in the SO(3) parametrization we have chosen). It is then automatic to find the

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JHEP11(2013)004

0 1 2 3 4 5 6 7 8 9

D3 x x x x

2 D5 x x x x x x

NS5 x x x x x x

D1 x x

Figure 8. The mirror ‘magnetic’ configuration, an external vortex in the theory of a free hyper-

multiplet.

0 1 2 3 4 5 6 7 8 9

D3 x x x x

2 NS5 x x x x x x

D5 x x x x x x

D1 x x

Figure 9. The brane configuration for a (localized) source of B and a (localized) FI term in N = 4

QED.

0 1 2 3 4 5 6 7 8 9

D3 x x x x

2 D5 x x x x x x

NS5 x x x x x x

F1 x x

Figure 10. The source for an ‘electric’ charge of U(1)J in the mirror theory.

magnetic dual — one simply applies type IIB S-duality to the brane configuration including

the source. The result is in figure 8.

Similar IIB brane configurations geometrizing the external magnetic defect in N = 4

QED, and its mirror electric source for U(1)J , are shown in figures 9 and 10. The insertion

of the semi-infinite D1 brane in the electric theory realizes the localized FI term introduced

in section 3.3. In the D-brane picture, the D1 brane pulls on one of the NS5 branes along

along its worldvolume direction 7, which is an FI term localized around the point where

the brane is inserted. (The same effect along the time direction gives rise to a localized

external A0.) As a limit of this, a smeared density of D1 branes would give rise to a uniform

displacement of an NS5 brane, which is the usual constant FI term.

In all cases, the semi-infinite F1s and D1s behave as external sources in the field theory

because of their infinite mass. Solving for the field configurations generated by these sources

(by finding bulk supergravity solutions incorporating ‘brane bending’) should reproduce the

direct field theory considerations of section 3. The N = 2 theories with external sources

can be engineered in the obvious similar manner, and we do not discuss them here.

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JHEP11(2013)004

8 Discussion and future directions

In this paper, we have started to explore the fact that defects in supersymmetric theories

can preserve half of the supersymmetry, giving rise to models of defects interacting with

a strongly coupled gauge theory that can be surprisingly tractable. This includes super-

symmetric models at finite density or in the presence of magnetic flux. There are many

directions for further exploration.

A crucial role in our analysis was played by the mirror symmetry of 3d supersymmet-

ric gauge theories, which in the particular case of 3d N = 4 QED maps questions about

defect interactions with the strongly coupled IR theory to dual questions in a free mag-

netic description. This allows one to write down simple explicit solutions reflecting the

backreaction of arbitrary arrays of defect charges on the theory, as in section 5. Further

exploration of these configurations, in particular of linear response and transport in the

presence of a defect lattice, could prove interesting.

Our analysis also suggested possible generalizations of the Abrikosov-Nielsen-Olesen

vortices to a spatially varying FI term and magnetic flux source, which would be very

interesting to analyze. We discussed only the simplest mirror pairs of theories with 3d

N = 4 and N = 2 supersymmetry. Much richer collections of mirror pairs are known [6–

11], and new phenomena may be visible there. Similarly, systems with Chern-Simons

terms are of considerable interest in condensed matter physics, and the exploration of

analogous defect configurations to the ones we discussed here in supersymmetric Chern-

Simons theories (for relatively comprehensive recent discussions, see [32, 33]) should be

straightforward.

We focused here on theories in 2+1 space-time dimensions. But the existence of super-

symmetry preserving defects is common to theories with BPS particles, including 4d N = 2

theories and various 2d supersymmetric theories. The key is that there should be a scalar

in the gauge multiplet that can cancel the SUSY-variation due to the finite charge density.

Exploring such defect models, perhaps using the tools of duality in other dimensions, is

likely to be worthwhile.

Acknowledgments

We would like to thank K. Intriligator, S. Sachdev and D. Tong for useful comments on

a draft of this work, and T. Cohen for collaboration on an early attempt at this analysis.

The research of S.K. and G.T. is supported in part by the National Science Foundation

under grant no. PHY-0756174. S.K. is also supported by the Department of Energy under

contract DE-AC02-76SF00515, and the John Templeton Foundation. A.H. is supported by

the Department of Energy under contract DE-SC0009988.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License which permits any use, distribution and reproduction in any medium,

provided the original author(s) and source are credited.

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JHEP11(2013)004

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